K-theory of split radical extensions

Recall that ifB is a ring, the Jacobson radicalrad(M)of aB-moduleM is the intersection of all the kernels of maps fromM to simple modules [10, p. 83]. Of particular importance to us is the case of a nilpotent idealI ⊆B. Then I ⊆rad(B) since1 +I consists of units.

We now turn to the very special task of giving a suitable model for K(B)whenf: B → Ais a split surjection with kernelIcontained in the Jacobson radicalrad(B)⊆B. We have some low dimensional knowledge about this situation, namely 1.2.3. and 1.3.2.5. which tell us that K0(B)∼=K0(A)and that the multiplicative group (1 +I)^× maps surjectively onto the kernel of the surjection K1(B) ։ K1(A). Some knowledge of K2 was also available already in the seventies (see e.g., [45] [227] and [144])

We use the strictly functorial model explained in 2.1.3 for the category of finitely generated projective modules P_A where an object is a pair (m, p) where m is a natural number and p∈MmA satisfiesp² =p. If j: A→B, then j∗(m, p) = (m, j(p)).

2. THE ALGEBRAIC K-THEORY SPECTRUM. 47 Lemma 2.5.1 Let f: B → A be a split surjective k-algebra map with kernel I, and let j: A → B be a splitting. Let c = (m, p) ∈ P_A and P = im(p), and consider P_B(j∗c, j∗c) as a monoid under composition. The kernel of the monoid map

f∗: PB(j∗c, j∗c)→ PA(c, c)

is isomorphic to the monoid of matrices x = 1 + y ∈ Mm(B) such that y ∈ MmI and y = yj(p) = j(p)y. This is also naturally isomorphic to the set M_A(P, P ⊗_Aj^∗I). The monoid structure induced on MA(P, P ⊗Aj^∗I) is given by

α·β = (1 +α)◦(1 +β)−1 =α+β+α◦β for α, β ∈ M_A(P, P ⊗_AI) where α◦β is the composite

P −−−→^β P ⊗_AI −−−→^α⊗1 P ⊗_AI⊗_AI multiplication inI

−−−−−−−−−−→ P ⊗_AI

Proof: As in definition 2.1.3, we identifyP_B(j∗c, j∗c)with the set of matrices x∈Mm(B) such that x=xj(p) =j(p)x and likewise for P_A(c, c). The kernel consists of the matrices x for which f(x) = p (the identity!), that is, the matrices of the form j(p) + y with y ∈ Mm(I) such that y = yj(p) = j(p)y. As a set, this is isomorphic to the claimed monoid, and the mapj(p) +y7→1 +yis a monoid isomorphism since (j(p) +y)(j(p) +z) = j(p)²+yj(p) +j(p)z+yz = j(p) +y+z +yz 7→ 1 +y+z+yz = (1 +y)(1 +z). The identification with MA(P, P ⊗Aj^∗I)is through the composite

HomA(P, P ⊗Aj^∗I) −−−→^∼= HomB(P ⊗AB, P ⊗AI)

φ7→1+φ

−−−−→ HomB(P ⊗_AB, P ⊗_AB) −−−→ P^∼= _B(j∗c, j∗c)

where the first isomorphism is the adjunction isomorphism and the last isomorphism is the natural isomorphism between

P_A −−−→ P^j∗ _B −−−→ M_B and

PA −−−→ MA

−⊗AB

−−−−→ MB.

Lemma 2.5.2 In the same situation as the preceding lemma, if I ⊂ Rad(B), then the kernel of

f∗: P_B(j∗c, j∗c) −−−→ P_A(c, c) is a group.

Proof: To see this, assume first that P ∼=Aⁿ. Then

MA(P, P ⊗AI)∼=MnI ⊆Mn(rad(B)) =rad(Mn(B))

(we have that Mn(rad(B)) = Rad(Mn(B)) since MB(Bⁿ,−) is an equivalence from B -modules to Mn(B)-modules, [10, p. 86]), and so (1 +Mn(I))^× is a group. If P is a direct summand of Aⁿ, say Aⁿ=P ⊕Q, andα ∈ M_A(P, P ⊗_AI), then we have a diagram

P ⊗AB −−−→^1+α P ⊗AB



y y Aⁿ⊗_AB −−−−→^1+(α,0) Aⁿ⊗_AB

where the vertical maps are split injections. By the discussion above1 + (α,0)must be an isomorphism, forcing1 +α to be one too.

All of the above holds true if instead of considering module categories, we consider the S construction of Waldhausen applied n times to the projective modules. More precisely, let nowcbe some object in Sp⁽ⁿ⁾P_A. Then the set of morphismsSp⁽ⁿ⁾M_A(c, c⊗_Aj^∗I)is still isomorphic to the monoid of elements sent to the identity under

Sp⁽ⁿ⁾P_B(j∗c, j∗c) −−−→^f∗ Sp⁽ⁿ⁾P_A(c, c)

and, if I is radical, this is a group. We will usually suppress the simplicial indices and speak of elements in some unspecified dimension. We will also usually suppress thej^∗ that should be inserted whenever I is considered as an A-module.

We need a few technical definitions.

Definition 2.5.3 Let

0 −−−→ I −−−→ B −−−→^f A −−−→ 0

be a split extension of k-algebras with I ⊂ Rad(B), and choose a splitting j: A → B of f. Let tP_B ⊆ P_B be the subcategory with all objects, but with morphisms only the endomorphisms taken to the identity by f∗. Note that, since I ⊆ rad(B), all morphisms in tP_B are automorphisms.

Let

tS_q⁽ⁿ⁾PB ⊆iS_q⁽ⁿ⁾PB

be the subcategory with the same objects, but with morphisms transformations of diagrams in Sq⁽ⁿ⁾P_B consisting of morphisms in tP_B.

Consider the sequence of (multi) simplicial exact categories n7→ D_AⁿB given by obD_AⁿB =obS⁽ⁿ⁾PA and D_AⁿB(c, d) = S⁽ⁿ⁾PB(j∗c, j∗d)

LettD_AⁿB ⊂ Dⁿ_ABbe the subcategory containing all objects, but whose only morphisms are the automorphismsS⁽ⁿ⁾MA(c, c⊗AI)considered as the subset{b ∈S⁽ⁿ⁾PB(j∗c, j∗c)|f∗b = 1} ⊆ D_AⁿB(c, c).

We set

K_AB ={n 7→BtD_AⁿB = a

m∈S⁽ⁿ⁾PA

B S⁽ⁿ⁾MA(m, m⊗AI)

} (2.5.3)

where the bar construction is taken with respect to the group structure.

2. THE ALGEBRAIC K-THEORY SPECTRUM. 49 Recall that in the eyes of K-theory there really is no difference between the special type of automorphisms coming from t and all isomorphisms since by corollary 2.3.2 the inclusions

obS⁽ⁿ⁾PB ⊆BtS⁽ⁿ⁾PB ⊆BiS⁽ⁿ⁾PB

are both weak equivalences.

Note thatD_AⁿB depends not only onI as anA-bimodule, but also on the multiplicative structure it inherits as an ideal in B. We have a factorization

S⁽ⁿ⁾P_A −−−→ D^{j! n}_AB −−−→^j# S⁽ⁿ⁾P_B

wherej! is the identity on object, andj∗ on morphisms, and j# is the fully faithful functor sending c∈ obtDⁿ_AB =obS⁽ⁿ⁾PA to j∗c∈ obS⁽ⁿ⁾PB (and the identity on morphisms). We see thatK_AB is a subspectrum of {n7→BiS⁽ⁿ⁾P_B} via

tD_AⁿB −−−→ tS⁽ⁿ⁾PB ⊆iS⁽ⁿ⁾PB

Theorem 2.5.4 Let f: B → A be a split surjection of k-algebras with splitting j and kernel I ⊂Rad(B). Then

Dⁿ_AB −−−→^j# S⁽ⁿ⁾PB, and its restriction tDⁿ_AB −−−→^j# tS⁽ⁿ⁾PB

are (degreewise) equivalences of simplicial exact categories, and so the chain K_AB(n) =BtDⁿ_AB ⊆BtS⁽ⁿ⁾P_B ⊇obS⁽ⁿ⁾P_B =K(B)(n) consists of weak equivalences.

Proof: To show that

D_AⁿB −−−→^j# S⁽ⁿ⁾P_B

is an equivalence, all we have to show is that every object in S⁽ⁿ⁾P_B is isomorphic to something in the image of j_#. We will show that c ∈ S⁽ⁿ⁾P_B is isomorphic to j∗f∗c = j#(j!f∗c).

Let c= (m, p)∈obP_B, P =im(p). Consider the diagram with short exact columns im(p)·I

 _

//____ im(jf(p))·I

 _

im(p)

π

ηp

//

_ _ _ _

_ im(jf(p))

π^′

f^∗im(f(p)) f^∗im(f jf(p))

Since im(p) is projective there exist a (not necessarily natural) lifting ηp. Let C be the cokernel of ηp. A quick diagram chase shows that C·I =C. Since im(jf(p)), and hence

C, is finitely generated, Nakayama’s lemma III.1.4.1 tells us thatC is trivial. This implies that ηp is surjective, but im(jf(p)) is also projective, so ηp must be split surjective. Call the splitting ǫ. Since πǫ = π^′η_pǫ = π^′ the argument above applied to ǫ shows that ǫ is also surjective. Hence ηp is an isomorphism. Thus, every objectc∈obPB is isomorphic to j∗(f∗c).

Let c ∈ obS⁽ⁿ⁾PB. Then c and j∗f∗c are splittable diagrams with isomorphic vertices.

Choosing isomorphisms on the “diagonal” we can extend these to the entire diagram, and so cand j∗f∗c are indeed isomorphic as claimed, proving the first assertion.

To show that

tD_AⁿB −−−→^j# tS⁽ⁿ⁾PB

is an equivalence, note first that this functor is also fully faithful. We know that any c ∈ ob tS⁽ⁿ⁾P_B = obS⁽ⁿ⁾P_B is isomorphic in S⁽ⁿ⁾P_B to j∗f∗c, and the only thing we need to show is that we can choose this isomorphism in t. Let ι: c → j∗f∗c∈ iS⁽ⁿ⁾PB be any isomorphism. Consider

c −−−→^ι j∗f∗c=j∗f∗j∗f∗c −−−−−→^{j∗f∗(ι−1)} j∗f∗c

Sincef∗(j∗f∗(ι⁻¹)◦ι) =f∗(ι⁻¹)◦f∗(ι) = 1f∗c the composite j∗f∗(ι⁻¹)◦ι is an isomorphism in tS_qⁿP fromc toj_#(j_!f∗c).

We set

Definition 2.5.5

K˜_AB =K_AB/K(A) ={n 7→ _

m∈S⁽ⁿ⁾PA

B S⁽ⁿ⁾M_A(m, m⊗_AI) }

Theorem 2.5.4 says that

K˜_AB −−−→^∼ K(B)/K(A)

is a (pointwise) equivalence of spectra. The latter spectrum is stably equivalent to the fiber ofK(B)→K(A). To see this, consider the square

K(B) −−−→ K(A)

 y

 y K(B)/K(A) −−−→ ∗

It is a (homotopy) cocartesian square of spectra, and hence homotopy cartesian. (In spectrum dimension n this is a cocartesian square, and the spaces involved are at least n−1-connected, and so all maps are n−1-connected. Then Blakers–Massey A.7.2.2 tells us that the square is (n−1) + (n −1)−1 = 2n−3 homotopy cartesian.) This means that the homotopy fiber of the upper horizontal map maps by a weak equivalence to the homotopy fiber of the lower horizontal map.

2. THE ALGEBRAIC K-THEORY SPECTRUM. 51 2.5.6 “Analyticity properties” of K_A(B)

The following result will not be called for until lemma VII.2.1.3, and may be safely skipped at a first reading until the result is eventually referred back to, but is placed here since it uses notation that is better kept local.

Although we are not using the notion of calculus of functors in these notes, we will in many cases come quite close. The next lemma, which shows how K_A(B) behaves under certain inverse limits, can be viewed as an example of this. A twist, which will reappear later is that we do not ask whether the functor turns “cocartesianness” into “cartesianness”, but rather to what extent the functor preserves inverse limits. The reason for this is that in many cases the coproduct structure of the source category can be rather messy, whereas some forgetful functor tells us exactly what the limits should be.

For the basics on cubes see appendix A.7. In particular, a strongly cocartesian n-cube is an n-cube where each two-dimensional face is cocartesian.

Let Split be the category of split radical extensions over a given ring A. The category sSplit of simplicial objects in Split then inherits the notion of k-cartesian cubes via the forgetful functor down to simplicial sets. By “final maps” in an n-cube we mean the maps induced from the n inclusions of the subsets of cardinality n−1in {1, . . . , n}.

If A⋉P ∈ sSplit it makes sense to talk about K(A⋉P) by applying the functor in every degree, and diagonalizing.

Lemma 2.5.7 Let A⋉P be a strongly cartesian n-cube in sSplit such all the final maps are k-connected. Then K(A⋉P) is (1 +k)n-cartesian.

Proof: Fix the non-negative integer q, the tuple p = (p1, . . . , pq) and the object c ∈ obSp^(q)PA. The cubeSp^(q)MA(c, c⊗AP)is also strongly cartesian (it is so as a simplicial set, and so as a simplicial group), and the final maps are stillk-connected. Taking the bar of this gives us a strongly cartesian cubeBSp^(q)MA(c, c⊗AP), but whose final maps will bek+ 1-connected. By the Blakers–Massey theorem A.7.2.2 this means that BSp^(q)MA(c, c⊗AP) will be (k+ 2)n−1-cocartesian. The same will be true for

a

c∈obSp^(q)PA

BS_p^(q)M_A(c, c⊗_AP)

Varying p and remembering that each multi-simplicial space is (q −1)-connected in the p direction, we see that the resulting cube is q+ (k+ 2)n−1-cocartesian, c.f. A.5.0.9.

Varying also q, we see that this means that the cube of spectra K(A⋉P) is(k+ 2)n−1 cocartesian, or equivalently (k+ 2)n−1−(n−1) = (k+ 1)n-cartesian.

The importance of this lemma will become apparent as we will approximate elements in Splitby means of cubical diagrams insSplitwhere all but the initial node will be “reduced”

in the sense that the zero skeletons will be exactly the trivial extension A=A.

In document The local structure of algebraic K-theory (sider 46-52)